We show that a flat metasurface with a parabolic reflection-phase distribution can focus an impinging plane wave to a point image in reflection geometry. Our system is much thinner than conventional geometric-optics devices and does not suffer the energy-loss issues encountered by many metamaterial devices working in transmission geometry. We designed realistic microwave samples and performed n...

A packet of light rays is defined that is equivalent to a Gaussian beam of light. The transformation of the Gaussian beam as it passes through any combination of perfect lenses and flat dielectric interfaces can be found by applying geometric optics to the equivalent ray packet.. The envelope of the ray packet gives the Gaussian beam spot size and the curves perpendicular to the average ray slo...

We study the geometry of a semiflexible polymer at finite temperatures. The writhe can be calculated from the properties of Gaussian random walks on the sphere. We calculate static and dynamic writhe correlation functions. The writhe of a polymer is analogous to geometric or Berry phases studied in optics and wave mechanics. Our results can be applied to confocal microscopy studies of stiff fil...

We study the flow map associated to the cubic Schrödinger equation in space dimension at least three. We consider initial data of arbitrary size in Hs, where 0 < s < sc, sc the critical index, and perturbations in Hσ , where σ < sc is independent of s. We show an instability mechanism in some Sobolev spaces of order smaller than s. The analysis relies on two features of super-critical geometric...

We experimentally investigate the focusing properties of a triangular two-dimensional photonic crystals of which all the cylinders are coated. The relationship between the location of object and image point is analyzed. It’s demonstrated that at a certain frequency with relative refractive index of −1, the position of image point obey the geometric-optics analysis and non-near-field imaging can...

The algebraic geometric approach to N -component systems of nonlinear integrable PDE’s is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical pheno...

We study the homogenization of an unsteady fluid-structure interaction problem with a scaling corresponding to a long time asymptotic regime. We consider oscillating initial data which are Bloch wave packets corresponding to tubes vibrating in opposition of phase. We prove that the initial displacements follow the rays of geometric optics and that the envelope function evolves according to a Sc...

We study the flow map associated to the cubic Schrödinger equation in space dimension at least three. We consider initial data of arbitrary size in Hs, where 0 < s < sc, sc the critical index, and perturbations in Hσ , where σ < sc is independent of s. We show an instability mechanism in some Sobolev spaces of order smaller than s. The analysis relies on two features of super-critical geometric...

Many problems in geometric optics or convex geometry can be recast as optimal transport problems and a popular way to solve these problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We introduce a damped Newton’s algorithm for this type of problems, which is experimentally efficient, and we establish its ...

The geometric optics stability method is extended to a general class of linear advective PDE’s with pseudodifferential bounded perturbation. We give a new short proof of Vishik’s formula for the essential spectral radius. We show that every point in the dynamical spectrum of the corresponding bicharacteristicamplitude system contributes a point into the essential spectrum of the PDE. Generic sp...